Reconfiguration in bounded bandwidth and treedepth

We show that several reconfiguration problems known to be PSPACE-complete remain so even when limited to graphs of bounded bandwidth. The essential step is noticing the similarity to very limited string rewriting systems, whose ability to directly simulate Turing Machines is classically known. This resolves a question posed open in [Bonsma P., 2012]. On the other hand, we show that a large class of reconfiguration problems becomes tractable on graphs of bounded treedepth, and that this result is in some sense tight.Comment: 14 page

Effectively Solving NP-SPEC Encodings by Translation to ASP

NP-SPEC is a language for specifying problems in NP in a declarative way. Despite the fact that the semantics of the language was given by referring to Datalog with circumscription, which is very close to ASP, so far the only existing implementations are by means of ECLiPSe Prolog and via Boolean satisfiability solvers. In this paper, we present translations from NP-SPEC into ASP, and provide an experimental evaluation of existing implementations and the proposed translations to ASP using various ASP solvers. The results show that translating to ASP clearly has an edge over the existing translation into SAT, which involves an intrinsic grounding process. We also argue that it might be useful to incorporate certain language constructs of NPSPEC into mainstream ASP

Recoloring bounded treewidth graphs

Let $k$ be an integer. Two vertex $k$-colorings of a graph are \emph{adjacent} if they differ on exactly one vertex. A graph is \emph{$k$-mixing} if any proper $k$-coloring can be transformed into any other through a sequence of adjacent proper $k$-colorings. Any graph is $(tw+2)$-mixing, where $tw$ is the treewidth of the graph (Cereceda 2006). We prove that the shortest sequence between any two $(tw+2)$-colorings is at most quadratic, a problem left open in Bonamy et al. (2012). Jerrum proved that any graph is $k$-mixing if $k$ is at least the maximum degree plus two. We improve Jerrum's bound using the grundy number, which is the worst number of colors in a greedy coloring.Comment: 11 pages, 5 figure

Recoloring graphs via tree decompositions

Let $k$ be an integer. Two vertex $k$-colorings of a graph are \emph{adjacent} if they differ on exactly one vertex. A graph is \emph{$k$-mixing} if any proper $k$-coloring can be transformed into any other through a sequence of adjacent proper $k$-colorings. Jerrum proved that any graph is $k$-mixing if $k$ is at least the maximum degree plus two. We first improve Jerrum's bound using the grundy number, which is the worst number of colors in a greedy coloring. Any graph is $(tw+2)$-mixing, where $tw$ is the treewidth of the graph (Cereceda 2006). We prove that the shortest sequence between any two $(tw+2)$-colorings is at most quadratic (which is optimal up to a constant factor), a problem left open in Bonamy et al. (2012). We also prove that given any two $(\chi(G)+1)$-colorings of a cograph (resp. distance-hereditary graph) $G$, we can find a linear (resp. quadratic) sequence between them. In both cases, the bounds cannot be improved by more than a constant factor for a fixed $\chi(G)$. The graph classes are also optimal in some sense: one of the smallest interesting superclass of distance-hereditary graphs corresponds to comparability graphs, for which no such property holds (even when relaxing the constraint on the length of the sequence). As for cographs, they are equivalently the graphs with no induced $P_4$, and there exist $P_5$-free graphs that admit no sequence between two of their $(\chi(G)+1)$-colorings. All the proofs are constructivist and lead to polynomial-time recoloring algorithmComment: 20 pages, 8 figures, partial results already presented in http://arxiv.org/abs/1302.348